If $\bar{a}=(2 \hat{i}+2 \hat{j}+3 \hat{k}), \bar{b}=(-\hat{i}+2 \hat{j}+\hat{k}) \quad$ and $\overline{\mathrm{c}}=(3 \hat{\mathrm{i}}+\hat{\mathrm{j}})$ such that $(\overline{\mathrm{a}}+\lambda \overline{\mathrm{b}})$ is perpendicular to $\bar{c}$, then the value of $\lambda$ is

Question

Aicharya · Accepted Answer

$(\overline{\mathrm{a}}+\lambda \overline{\mathrm{b}})=(2-\lambda) \hat{\mathrm{i}}+(2+2 \lambda) \hat{\mathrm{j}}+(3+\lambda) \hat{\mathrm{k}}$
Since $\bar{a}+\lambda \bar{b}$ is perpendicular to $\bar{c}$ $\begin{aligned} \therefore \quad & (\bar{a}+\lambda \bar{b}) \cdot \bar{c}=0 \\ & \Rightarrow(2-\lambda)(3)+(2+2 \lambda)(1)+(3+\lambda)(0)=0 \\ & \Rightarrow 6-3 \lambda+2+2 \lambda=0 \\ & \Rightarrow \lambda=8 \end{aligned}$

If $\bar{a}=(2 \hat{i}+2 \hat{j}+3 \hat{k}), \bar{b}=(-\hat{i}+2 \hat{j}+\hat{k}) \quad$ and…

Solution